A different view of independent sets in bipartite graphs
Qi GeDaniel Štefankovič
University of Rochester
counting/sampling independent sets in general graphs:
A different view of independent sets in bipartite graphs
polynomial time sampler for 5 (Dyer,Greenhill ’00, Luby,Vigoda’99, Weitz’06).
no polynomial time sampler (unless NP=RP) for 25 (Dyer, Frieze, Jerrum ’02).
Glauber dynamics does not mix in polynomial time for 6-regular bipartite graphs (example: union of 6 random matchings) (Dyer, Frieze, Jerrum ’02).
= maximum degree of G
counting/sampling independent sets in bipartite graphs:
A different view of independent sets in bipartite graphs
polynomial time sampler for 5 (Dyer,Greenhill ’00, Luby,Vigoda’99, Weitz’06).
no polynomial time sampler (unless NP=RP) for 25 (Dyer, Frieze, Jerrum ’02).
Glauber dynamics does not mix in polynomial time for 6-regular bipartite graphs (example: union of 6 random matchings) (Dyer, Frieze, Jerrum ’02).
(max idependent set in bipartite graph max matching)
= maximum degree of G
Why do we care?
How hard is counting/sampling independent sets in bipartite graphs?
* bipartite independent sets
equivalent to
* enumerating solutions of a linear Datalog program * downsets in a poset (Dyer, Goldberg, Greenhill, Jerrum’03) * ferromagnetic Ising with mixed external field (Goldberg,Jerrum’07) * stable matchings (Chebolu, Goldberg, Martin’10)
0 00 0
0 00 1
1 01 0
1 01 1
0 10 0
0 01 0
1 11 1
0 11 1
1 00 0
0 01 1
0 11 0
1 10 1
1 10 0
0 10 1
1 00 1
1 11 0
Independent sets in a bipartite graph. 0-1 matrices weighted by
(1/2)rank (1 allowed at Auv if uv is an edge)
Ge, Štefankovič ’09
A different view of independent sets in bipartite graphs
0 00 0
1 01 0
0 10 0
0 01 0
1 00 0
0 11 0
1 10 0
1 11 0
A different view of independent sets in bipartite graphs
Ge, Štefankovič ’09
#IS = 2|VU| - |E| 2-rk(A)
A B
Independent sets in a bipartite graph. 0-1 matrices weighted by
(1/2)rank (1 allowed at Auv if uv is an edge)
0 00 0
1 01 0
0 10 0
0 01 0
1 00 0
0 11 0
1 10 0
1 11 0
A different view of independent sets in bipartite graphs
Ge, Štefankovič ’09
#IS = 2|V U| - |E| 2-rk(A)
A B
Independent sets in a bipartite graph. 0-1 matrices weighted by
(1/2)rank (1 allowed at Auv if uv is an edge)
Question: Is there a polynomial-time samplerthat produces matrices A B with P(A) 2-rank(A)
Bij=0 Aij=0
(everything over the F2)
Natural MC
flip random entry + Metropolis filter.
A = Xt with random (valid) entry flipped
if rank(A) rank(Xt) then Xt+1 = A if rank(A) > rank(Xt) then Xt+1 = A w.p. ½ Xt+1 = Xt w.p. ½
we conjectured it is mixing
Goldberg,Jerrum’10: the chain is exponentially slow for some graphs.
BAD NEWS:
Ising model: assignment of spinsto sites weighted by the numberof neighbors that agree
Random cluster model: subgraphsweighted by the number of components and the number ofedges
High temperature expansion: even subgraphs weightedby the number of edges
Our inspiration (Ising model):
Fortuin-Kasteleyn
Newell M
ontroll ‘53
Random cluster model
Z(G,q,)= q(S)|S|
SE
number of connectedcomponents of (G,S)
(Tutte polynomial)Ising modelPotts modelchromatic polynomialFlow polynomial
Random cluster model
Z(G,q,)= q(S)|S|
SE
R2 model
R2(G,q,)= qrk(S)|S|
SE2
number of connectedcomponents of (G,S)
rank (over F2) of theadjacency matrix of (G,S)
(Tutte polynomial)Ising modelPotts modelchromatic polynomialFlow polynomial
MatchingsPerfect matchings
Independent sets (for bipartite only!)
More ?
R2 model’
easy if (x-1)(y-1)=1, or(1,1),(-1,-1),(0,-1),(-1,0)
#P-hard elsewhere
Tutte polynomial
easy if q{0,1} or =0, or (1/2,-1)
#P-hard elsewhere (GRH)
Complexity of exact evaluation
Ge, Štefankovič ’09Jaeger, Vertigan, Welsh ’90
2|E|-|V|+|isolated V|
spanning trees
R2(G,q,)= qrk (S)|S|
SE2‘
BIS
q
“high-temperature expansion”
(1-((u),(v))U{0,1} V{0,1} {u,v}E
1,1) = 10,1) = (1,0) = (0,0) = -1
where
2|E| #BIS =
“high-temperature expansion”
(1-((u),(v))U{0,1} V{0,1} {u,v}E
1,1) = 10,1) = (1,0) = (0,0) = -1
where
2|E| #BIS =
= (-1)|S| ((u),(v))SE U{0,1} V{0,1} {u,v}S
“high-temperature expansion”
(1-((u),(v))U{0,1} V{0,1} {u,v}E
1,1) = 10,1) = (1,0) = (0,0) = -1
where
2|E| #BIS =
= (-1)|S| ((u),(v))SE U{0,1} V{0,1} {u,v}S
= {0 if some v V has an odd number of neighbors in (UV,S) labeled by 1(-2)|V| otherwise
“high-temperature expansion”
2|E| #BIS =
= (-1)|S| ((u),(v))SE U{0,1} V{0,1} {u,v}S
bipartite adjacency matrix of (UV,S)
= 2|V|SE
number of u such that uTA = 0 (mod 2)
= 2|V|+|U| SE
2- rank (A))2
“high-temperature expansion” – curious
f(A,) = |v| ( )|Av| 1-1+
f(A,1) = 2rank (A)
1 1
2
f(A,1) = f(A,1)T
But in fact:
f(A,) = f(A,)T
Questions:
Is there a polynomial-time sampler that produces matrices A B with P(A) 2-rank(A) ?
What other quantities does the R2 polynomial encode ?
R2(G,q,)= qrk(S)|S|
SE2